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Title: The join construction for free involutions on spheres
Author: Macko, Tibor
ISNI:       0000 0001 3615 6566
Awarding Body: University of Aberdeen
Current Institution: University of Aberdeen
Date of Award: 2003
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Given a Poincaré complex X, the set of manifold structures on X x Dk relative to the boundary can be viewed as the k-th homotopy group of a space Ss(X). This space is called the block structure space of X. Free involutions on spheres are in one-to-one correspondence with manifold structures on real projective spaces.  There is Wall’s join construction for free involutions on spheres which we generalize to a map between block structure spaces of real projective spaces.  More precisely, we define a functor from the category of real finite-dimensional vector spaces with inner product to pointed spaces which to a vector space V assigns the block structure space of the projective space of V.  We study this functor from the point of view of orthogonal calculus of functors, we prove that the 6-fold delooping of the first derivative spectrum of this functor is an W-spectrum. The proof uses mainly codimension 1 surgery theory. This results suggests an attractive description of the block structure space of the infinite-dimensional real projective space which is a colimit of block structure spaces of projective spaces of finite-dimensional real vector spaces. The description is via the Taylor tower of orthogonal calculus.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available